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Mirrors > Home > ILE Home > Th. List > disjsn | Unicode version |
Description: Intersection with the singleton of a non-member is disjoint. (Contributed by NM, 22-May-1998.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.) |
Ref | Expression |
---|---|
disjsn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disj1 3413 | . 2 | |
2 | con2b 658 | . . . 4 | |
3 | velsn 3544 | . . . . 5 | |
4 | 3 | imbi1i 237 | . . . 4 |
5 | imnan 679 | . . . 4 | |
6 | 2, 4, 5 | 3bitri 205 | . . 3 |
7 | 6 | albii 1446 | . 2 |
8 | alnex 1475 | . . 3 | |
9 | df-clel 2135 | . . 3 | |
10 | 8, 9 | xchbinxr 672 | . 2 |
11 | 1, 7, 10 | 3bitri 205 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wal 1329 wceq 1331 wex 1468 wcel 1480 cin 3070 c0 3363 csn 3527 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-v 2688 df-dif 3073 df-in 3077 df-nul 3364 df-sn 3533 |
This theorem is referenced by: disjsn2 3586 ssdifsn 3651 orddisj 4461 ndmima 4916 funtpg 5174 fnunsn 5230 ressnop0 5601 ftpg 5604 fsnunf 5620 fsnunfv 5621 enpr2d 6711 phpm 6759 fiunsnnn 6775 ac6sfi 6792 unsnfi 6807 tpfidisj 6816 iunfidisj 6834 pm54.43 7046 dju1en 7069 fzpreddisj 9851 fzp1disj 9860 frecfzennn 10199 hashunsng 10553 hashxp 10572 fsumsplitsn 11179 sumtp 11183 fsumsplitsnun 11188 fsum2dlemstep 11203 fsumconst 11223 fsumabs 11234 fsumiun 11246 ennnfonelemhf1o 11926 structcnvcnv 11975 fsumcncntop 12725 |
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